Question

in the coordinate plane, points a, b, and c have coordinates (1,2), (4,2), and (4, -1) respectively. 1. plot points a, b, and c on a coordinate grid, then connect them to form a figure. what is the shape of this figure? 2. calculate the area of the figure formed by points a, b, and c. 3. find the coordinates of point d such that quadrilateral abcd is a rectangle. plot point d and verify the rectangle by checking the properties of its sides.

Explanation:

Sub - Question 1

To determine the shape, we analyze the coordinates. For points \(A(1,2)\), \(B(4,2)\), and \(C(4, - 1)\):

• The slope of \(AB\): Using the slope formula \(m=\frac{y_2 - y_1}{x_2 - x_1}\), for \(A(1,2)\) and \(B(4,2)\), \(m_{AB}=\frac{2 - 2}{4 - 1}=0\), so \(AB\) is horizontal. The length of \(AB\) is \(|4 - 1| = 3\).
• The slope of \(BC\): For \(B(4,2)\) and \(C(4,-1)\), \(m_{BC}=\frac{-1 - 2}{4 - 4}\) is undefined, so \(BC\) is vertical. The length of \(BC\) is \(|2-(-1)| = 3\).
• The angle between \(AB\) (horizontal) and \(BC\) (vertical) is \(90^{\circ}\). And we have three points, so the figure is a right triangle.

Step 1: Identify the base and height of the right triangle

In a right triangle, the two legs can be considered as the base and the height. From the coordinates, the length of \(AB\) (horizontal leg) is \(4 - 1=3\) and the length of \(BC\) (vertical leg) is \(2-(-1) = 3\).

Step 2: Use the area formula for a right triangle

The area formula for a right triangle is \(A=\frac{1}{2}\times\text{base}\times\text{height}\). Here, base \(b = 3\) and height \(h=3\). So \(A=\frac{1}{2}\times3\times3\).

Step 3: Calculate the area

\(\frac{1}{2}\times3\times3=\frac{9}{2} = 4.5\)

Step 1: Recall the properties of a rectangle

In a rectangle, opposite sides are equal and parallel. For quadrilateral \(ABCD\) to be a rectangle, \(\overrightarrow{AB}=\overrightarrow{DC}\) and \(\overrightarrow{AD}=\overrightarrow{BC}\). We know \(A(1,2)\), \(B(4,2)\), \(C(4,-1)\). Let \(D(x,y)\).

• Since \(\overrightarrow{AB}=(4 - 1,2 - 2)=(3,0)\), and \(\overrightarrow{DC}=(4 - x,-1 - y)\), we have \(4 - x = 3\) and \(-1 - y=0\). From \(4 - x = 3\), we get \(x = 1\). From \(-1 - y = 0\), we get \(y=-1\)? Wait, no. Wait, \(\overrightarrow{AD}=\overrightarrow{BC}\). \(\overrightarrow{BC}=(4 - 4,-1 - 2)=(0,-3)\), and \(\overrightarrow{AD}=(x - 1,y - 2)\). So \(x - 1=0\) (so \(x = 1\)) and \(y - 2=-3\) (so \(y=-1\))? Wait, no, let's do it another way.

The vector from \(A\) to \(B\) is \((3,0)\), and the vector from \(B\) to \(C\) is \((0,-3)\). To get from \(C\) to \(D\), we need a vector equal to \(\overrightarrow{AB}\) but in the reverse direction of \(\overrightarrow{BC}\)? Wait, in a rectangle \(ABCD\), the coordinates of \(D\) can be found by \(D = A+(C - B)\). \(C - B=(4 - 4,-1 - 2)=(0,-3)\), so \(D=(1,2)+(0,-3)=(1,-1)\).

Step 2: Verify the properties

• Check the sides:
• \(AB\): from \((1,2)\) to \((4,2)\), length \(3\), horizontal.
• \(BC\): from \((4,2)\) to \((4,-1)\), length \(3\), vertical.
• \(CD\): from \((4,-1)\) to \((1,-1)\), length \(3\), horizontal (slope \(0\)).
• \(DA\): from \((1,-1)\) to \((1,2)\), length \(3\), vertical (slope undefined).
• Opposite sides are equal and parallel, and adjacent sides are perpendicular (horizontal and vertical lines are perpendicular). So \(ABCD\) is a rectangle.

Answer:

The shape of the figure is a right triangle.

Sub - Question 2